Pupil factor and linear perspective

[erikV]

Hi Littlefield

I somehow got back to this question now after almost a year. You helped me understand a lot back then. Now I am still not sure how to compare the final cropped-and-enlarged photoA to the original photoB, especially regarding DOF. Sometimes I even ask myself if this question is well-defined. I mean we are manipulating a finished photo, so how can we know if this reflects how some lens with a certain F-value will display the true 3D scene?

Here is a short resumé: Two photos are taken from the same position and in the same direction towards a 3D-scene. PhotoA is taken with a lens having focal length f_A and photoB is taken with a lens having a bigger focal length f_B. Now photoA is being cropped to cover the same framing as photoB and subsequently enlarged to have the same size as photoB. Let's call the cropped and enlarged photoA for photoA'. When regarding linear perspective photoA' and photoB are the same. This has been settled. The enlargement will be given by the scaling factor f_B/f_A (when the focal lengths are small compared to the object distance). This also ensures that the correct viewing distance, which is "attached" to every final photo, will be the same for photoA' and photoB. Formula:

Viewing distance = f*(photo dimension/sensor dimension)



So it all matches up when regarding perspective. But what about the blurriness? Let's assume the two photos are shot with the same aperture (F-number). Given the object distance and circle of confusion c, one can theoretically calculate the DOF using the formulas in the excellent article of Jeff Conrad linked to below. Generally speaking photos shot with a wideangle lens will show greater DOF compared to a photo shot with a lens with bigger focal length (everything else regarded the same). So photoA should display greater DOF than photoB. Now when photoA is resized and moved further away from the eye to have the new correct viewing distance, which is equal to the viewing distance for photoB, the blurriness should look like the same when being view from the eye's perspective. The angles are preserved! So with these arguments, photoA' should have the same DOF field as photoA and consequently a bigger DOF than photoB. What is wrong in this line of arguments? (I know I haven't taken the reduced pixelcount into consideration).

I expect photoB to have at least the same DOF as photoA' and probably less noise. I am pretty sure you can reveal what is wrong in my arguments regarding blurriness

Article mentioned: Depth of Field in Depth:
https://www.largeformatphotography.info ... nDepth.pdf

Regards,

Erik

[rjlittlefield]

Erik, welcome back!

Let me start by reformulating the question like this:

I want to shoot the same picture using two different size sensors. To accomplish that, what do I have to hold constant and what do I have to change?

For this question, a line of thought called "equivalent images analysis" gives the following answer:

• Hold constant the entrance pupil location and diameter.
• Hold constant the exposure time.
• Use longer focal length on the larger sensor, as needed to give the same framing.
• Increase ISO setting for the larger sensor, in proportion to the sensor area.
• Hold constant the total number of pixels on sensor, the sensors' quantum capture efficiency, and the sensors' per-pixel electronic noise.
• Enlarge both images to the same final size, and view them from the same location.

It turns out that this combination will give you

• same perspective,
• same DOF,
• same motion blur,
• same diffraction blur, and
• same image noise.

In other words, the resulting images will be identical except for random variations in the noise details.

I can summarize the whole issue by saying "If you capture the same light, you get the same image."

Getting back to your original question, an unexpected simplification that drops out of equivalent images analysis is that effective F-number, as seen by the sensor, has to scale exactly in proportion to the sensor's linear dimension. This holds under all conditions, even macro/micro work. There is no corresponding simplification for marked F-number as set on the lens. Both the focal length and the marked F-number may have to scale in complicated ways for finite focal distances. When focus is at infinity, effective F-number = marked F-number so it is sufficient to scale marked F-number in proportion to focal length, for example 50 mm f/4 versus 100 mm f/8, entrance pupil diameter = 12.5 mm for both.

At this point, probably the best value for both of us is if I turn you loose to go read about equivalent images analysis and figure out how this all fits together.

Unfortunately a lot of vague/incorrect/misleading stuff has been written in this area, so randomly searching the literature is more likely to get you hurt than helped.

One excellent exception that I highly recommend is http://www.josephjamesphotography.com/equivalence/index.htm . It is both comprehensive and very long, but fortunately it has a bullet-point summary up front.

Quite some years ago (2008) I had a long discussion here at PMN on the same topic (and referencing the article linked above). That discussion is at viewtopic.php?t=4108 . Unfortunately a good bit of the early discussion was trying to help another fellow get past some serious misconceptions, which he was posting as objections. I highly recommend reading the thread, but concentrating on my content more than the other fellow's.

If you have questions, please post them here rather than resurrecting the old thread.

--Rik

[erikV]

Hi again Littlefield. I wish you a happy new year!

Sorry for being so slow to respond. I was called out to some extra work before Christmas and because you gave me some considerable "homework" to do, I had to postpone it to the Christmas Holidays I have now read part of the long article about Equivalent photos on Joseph James Photography homepage. It is a comprehensive article indeed and it seems to have many well-thought out points. One drawback is however that the presentation is containing many words and around 100 pages is making it hard to get an overview and not get lost in details. Good with the examples, though. Unfortunately they are written with 1-dimensional math notation, making it harder to read.

I see your point now that my problem with cropping and scaling amounts to the same as to using a smaller sensor, namely the part of the whole sensor that gives rise to the part of the image that is left after cropping. Once I made an experiment with a zoom lens on a tripod. I took two photos, one with focal length 16 mm and one with focal length 68 mm. I did not think about aperture and ISO back then, because I was only interested in demonstrating that the focal length does not influence the perspective, only the angle which is displayed. Now here is what I calculated in Maple using the notation from the article of Rowlands (here):



I hope this is correct!

I assume that since it is the same zoom-lens being used when shooting the two photos, the noise will be very much the same (no difference in sensor effectivity).

Regards,

Erik

[rjlittlefield]

I hope this is correct!

The math looks correct to me.

Deep in the fine print, there is an issue that zooming a lens usually causes its entrance pupil to move forward or backward. If you were working at close distances this moving of the entrance pupil could affect the perspective enough to be detected by close study. For a test at landscape distance, it will have no visible effect.

--Rik

[erikV]

Yes the formulas don't hold for small focusing distances, which is often used in Macro Photography. Rowlands handles this situation too. I am however satisfied with the approximative results: These formulas are much easier to remember and a rule of thumb.

One thing that puzzles me a bit is the fact, that in the definition og Equivalence it is NOT required that the two sensor formats have the same ratio between length and width. It becomes a bit awkward to talk about the same framing then. In that situations the diagonals are being used to compare sensorsizes. This also makes it possible to compare for example a mobile phone sensor, which is usually 4:3 and a Full frame sensor, which is 3:2. Of course one could use the cropping argument as above to compare them even with a more strict definition ...

Erik

[rjlittlefield]

One thing that puzzles me a bit is the fact, that in the definition og Equivalence it is NOT required that the two sensor formats have the same ratio between length and width.
...
Of course one could use the cropping argument as above to compare them even with a more strict definition ...

Yep. In the phrasing of Joseph James' article at "Definition of Equivalence":

In addition, there is a small niggle in the parameter of "same framing" for systems with different aspect ratios (e.g. 4:3 vs 3:2). We can either crop one image to the the aspect ratio of the other (or crop both to a common aspect ratio) or compare at the same AOV and display with the same diagonal measurement. The details of this are discussed at the end of this section.

--Rik